Fractional Calculus and Special Functions with Applications
The study of fractional integrals and fractional derivatives has a long history, and they have many real-world applications because of their properties of interpolation between integer-order operators. This field includes classical fractional operators such as Riemann–Liouville, Weyl, Caputo, and Gr...
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| フォーマット: | Online |
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| 言語: | 英語 |
| 出版事項: |
MDPI - Multidisciplinary Digital Publishing Institute
2022
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| オンライン･アクセス: | ONIX_20220506_9783036536170_41 |
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| _version_ | 1863735799939334144 |
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| collection | Directory of Open Access Books |
| description | The study of fractional integrals and fractional derivatives has a long history, and they have many real-world applications because of their properties of interpolation between integer-order operators. This field includes classical fractional operators such as Riemann–Liouville, Weyl, Caputo, and Grunwald–Letnikov; nevertheless, especially in the last two decades, many new operators have also appeared that often define using integrals with special functions in the kernel, such as Atangana–Baleanu, Prabhakar, Marichev–Saigo–Maeda, and the tempered fractional equation, as well as their extended or multivariable forms. These have been intensively studied because they can also be useful in modelling and analysing real-world processes, due to their different properties and behaviours from those of the classical cases.Special functions, such as Mittag–Leffler functions, hypergeometric functions, Fox's H-functions, Wright functions, and Bessel and hyper-Bessel functions, also have important connections with fractional calculus. Some of them, such as the Mittag–Leffler function and its generalisations, appear naturally as solutions of fractional differential equations. Furthermore, many interesting relationships between different special functions are found by using the operators of fractional calculus. Certain special functions have also been applied to analyse the qualitative properties of fractional differential equations, e.g., the concept of Mittag–Leffler stability.The aim of this reprint is to explore and highlight the diverse connections between fractional calculus and special functions, and their associated applications. |
| format | Online |
| id | doab-20.500.12854ir-80975 |
| institution | Directory of Open Access Books |
| language | eng |
| publishDate | 2022 |
| publishDateRange | 2022 |
| publishDateSort | 2022 |
| publisher | MDPI - Multidisciplinary Digital Publishing Institute |
| publisherStr | MDPI - Multidisciplinary Digital Publishing Institute |
| record_format | ojs |
| spelling | doab-20.500.12854ir-809752024-03-28T03:32:19Z Fractional Calculus and Special Functions with Applications Özarslan, Mehmet Ali Fernandez, Arran Area, Ivan Caputo-Hadamard fractional derivative coupled system Hadamard fractional integral boundary conditions existence fixed point theorem fractional Langevin equations existence and uniqueness solution fractional derivatives and integrals stochastic processes calculus of variations Mittag-Leffler functions Prabhakar fractional calculus Atangana–Baleanu fractional calculus complex integrals analytic continuation k-gamma function k-beta function Pochhammer symbol hypergeometric function Appell functions integral representation reduction and transformation formula fractional derivative generating function physical problems fractional derivatives fractional modeling real-world problems electrical circuits fractional differential equations fixed point theory Atangana–Baleanu derivative mobile phone worms fractional integrals Abel equations Laplace transforms mixed partial derivatives second Chebyshev wavelet system of Volterra–Fredholm integro-differential equations fractional-order Caputo derivative operator fractional-order Riemann–Liouville integral operator error bound n/a thema EDItEUR::G Reference, Information and Interdisciplinary subjects::GP Research and information: general thema EDItEUR::P Mathematics and Science The study of fractional integrals and fractional derivatives has a long history, and they have many real-world applications because of their properties of interpolation between integer-order operators. This field includes classical fractional operators such as Riemann–Liouville, Weyl, Caputo, and Grunwald–Letnikov; nevertheless, especially in the last two decades, many new operators have also appeared that often define using integrals with special functions in the kernel, such as Atangana–Baleanu, Prabhakar, Marichev–Saigo–Maeda, and the tempered fractional equation, as well as their extended or multivariable forms. These have been intensively studied because they can also be useful in modelling and analysing real-world processes, due to their different properties and behaviours from those of the classical cases.Special functions, such as Mittag–Leffler functions, hypergeometric functions, Fox's H-functions, Wright functions, and Bessel and hyper-Bessel functions, also have important connections with fractional calculus. Some of them, such as the Mittag–Leffler function and its generalisations, appear naturally as solutions of fractional differential equations. Furthermore, many interesting relationships between different special functions are found by using the operators of fractional calculus. Certain special functions have also been applied to analyse the qualitative properties of fractional differential equations, e.g., the concept of Mittag–Leffler stability.The aim of this reprint is to explore and highlight the diverse connections between fractional calculus and special functions, and their associated applications. 2022-05-06T11:19:18Z 2022-05-06T11:19:18Z 2022 book ONIX_20220506_9783036536170_41 9783036536170 9783036536187 https://directory.doabooks.org/handle/20.500.12854/80975 eng image/jpeg Attribution 4.0 International https://mdpi.com/books/pdfview/book/5315 https://mdpi.com/books/pdfview/book/5315 MDPI - Multidisciplinary Digital Publishing Institute 10.3390/books978-3-0365-3618-7 10.3390/books978-3-0365-3618-7 46cabcaa-dd94-4bfe-87b4-55023c1b36d0 9783036536170 9783036536187 164 Basel open access |
| spellingShingle | Caputo-Hadamard fractional derivative coupled system Hadamard fractional integral boundary conditions existence fixed point theorem fractional Langevin equations existence and uniqueness solution fractional derivatives and integrals stochastic processes calculus of variations Mittag-Leffler functions Prabhakar fractional calculus Atangana–Baleanu fractional calculus complex integrals analytic continuation k-gamma function k-beta function Pochhammer symbol hypergeometric function Appell functions integral representation reduction and transformation formula fractional derivative generating function physical problems fractional derivatives fractional modeling real-world problems electrical circuits fractional differential equations fixed point theory Atangana–Baleanu derivative mobile phone worms fractional integrals Abel equations Laplace transforms mixed partial derivatives second Chebyshev wavelet system of Volterra–Fredholm integro-differential equations fractional-order Caputo derivative operator fractional-order Riemann–Liouville integral operator error bound n/a thema EDItEUR::G Reference, Information and Interdisciplinary subjects::GP Research and information: general thema EDItEUR::P Mathematics and Science Fractional Calculus and Special Functions with Applications |
| title | Fractional Calculus and Special Functions with Applications |
| title_full | Fractional Calculus and Special Functions with Applications |
| title_fullStr | Fractional Calculus and Special Functions with Applications |
| title_full_unstemmed | Fractional Calculus and Special Functions with Applications |
| title_short | Fractional Calculus and Special Functions with Applications |
| title_sort | fractional calculus and special functions with applications |
| topic | Caputo-Hadamard fractional derivative coupled system Hadamard fractional integral boundary conditions existence fixed point theorem fractional Langevin equations existence and uniqueness solution fractional derivatives and integrals stochastic processes calculus of variations Mittag-Leffler functions Prabhakar fractional calculus Atangana–Baleanu fractional calculus complex integrals analytic continuation k-gamma function k-beta function Pochhammer symbol hypergeometric function Appell functions integral representation reduction and transformation formula fractional derivative generating function physical problems fractional derivatives fractional modeling real-world problems electrical circuits fractional differential equations fixed point theory Atangana–Baleanu derivative mobile phone worms fractional integrals Abel equations Laplace transforms mixed partial derivatives second Chebyshev wavelet system of Volterra–Fredholm integro-differential equations fractional-order Caputo derivative operator fractional-order Riemann–Liouville integral operator error bound n/a thema EDItEUR::G Reference, Information and Interdisciplinary subjects::GP Research and information: general thema EDItEUR::P Mathematics and Science |
| topic_facet | Caputo-Hadamard fractional derivative coupled system Hadamard fractional integral boundary conditions existence fixed point theorem fractional Langevin equations existence and uniqueness solution fractional derivatives and integrals stochastic processes calculus of variations Mittag-Leffler functions Prabhakar fractional calculus Atangana–Baleanu fractional calculus complex integrals analytic continuation k-gamma function k-beta function Pochhammer symbol hypergeometric function Appell functions integral representation reduction and transformation formula fractional derivative generating function physical problems fractional derivatives fractional modeling real-world problems electrical circuits fractional differential equations fixed point theory Atangana–Baleanu derivative mobile phone worms fractional integrals Abel equations Laplace transforms mixed partial derivatives second Chebyshev wavelet system of Volterra–Fredholm integro-differential equations fractional-order Caputo derivative operator fractional-order Riemann–Liouville integral operator error bound n/a thema EDItEUR::G Reference, Information and Interdisciplinary subjects::GP Research and information: general thema EDItEUR::P Mathematics and Science |
| url | ONIX_20220506_9783036536170_41 |